In the art of making paper with modern high-speed machines, sheet properties must be continually monitored and controlled to assure sheet quality and to minimize the amount of finished product that is rejected when there is an upset in the manufacturing process. The sheet variables that are most often measured include basis weight, moisture content, and caliper (i.e., thickness) of the sheets at various stages in the manufacturing process. These process variables are typically controlled by, for example, adjusting the feedstock supply rate at the beginning of the process, regulating the amount of steam applied to the paper near the middle of the process, or varying the nip pressure between calendaring rollers at the end of the process.
A paper machine employs large arrays of actuators spread across a continuously moving web to control the cross-directional (CD) profiles of paper properties as measured by one (or several) scanning sensor(s) downstream from the actuators. Typically, designers are using pairing rules to choose one CD actuator array for controlling one paper sheet property and the interaction of multiple array CD processes is usually neglected in traditional CD control.
Most well-designed single array CD systems are unfortunately ill-conditioned. Even at steady-state, some of their singular values are vanishingly small. The large dimensionality and the ill-conditioning make these processes challenging to control. It has been recently suggested that for multiple array CD processes the ill-conditioning of the process could be due to the interaction between multiple array measurements and actuators. That means it can be more difficult to control multiple array CD systems than single array CD systems.
Application of model predictive control (MPC) in CD processes has been considered for some time. Although most published papers consider only one actuator array and one controlled property and consequently do not address the problem of coordinating multiple CD actuator arrays controlling multiple sheet properties, multiple array CD control systems are becoming more prevalent. Industrial model predictive control implementation can employ a multiple-array model of the CD process that is obtained from a complementary industrial model identification tool. The advantages of multiple-array control are evident in the improved performances that have been reported. The main disadvantage of online optimization is the enormous computational load required as the constrained quadratic programming (QP) problem may be required to generate as many as 600 actuator setpoints subject to up to 1800 constraints from up to 6000 measurements as often as every 15 seconds. The optimization problem is highly structured and optimization algorithms which exploit this structure have been developed. A potentially complementary technique is to use model reduction techniques to reduce the size of the optimization problem.
In R. Shridhar and D. J. Cooper, “A tuning strategy for unconstrained multivariable model predictive control,” Industrial & Engineering Chemistry & Research, vol. 37, no. 10, pp 4003-4016, 1998 and D. Dougherty and D. J. Cooper, “Tuning guidelines of a dynamic matrix controller for integrating (non-self-regulating) processes,” Industrial & Engineering Chemistry & Research, vol. 42, no. 8, pp 1739-1752, 2003, the authors proposed some tuning guidelines for multivariable dynamic matrix controllers. In K. Y. Rani and H. Unbehauen, “Study of predictive controller tuning methods,” Automatica, vol. 33, no 12, pp 2243-2248, 1997, the authors proposed tuning procedures for predictive controllers that are based on some tuning rules and closed-loop simulations. In J. H. Lee and Z. Yu, “Tuning of model predictive controllers for robust performance,” Computers & Chemical Engineering, vol. 18, no. 1, pp. 15-37, 1994, tuning rules based on the frequency-domain analysis of the closed-loop behavior of MPC controllers are presented. In A. Al-Ghazzawi, et al., “On-line tuning strategy for model predictive controllers,” Journal of Process Control, vol. 11, no. 3, pp. 265-284, 2001, an on-line tuning strategy for linear model predictive control algorithms is proposed based on the linear approximation between the closed-loop predicted output and the MPC tuning parameters. J. Trierweiler and L. A. Farina, “RPN tuning strategy for model predictive control,” Journal of Process Control, vol. 13, no. 7, pp. 591-598, 2003, presented a tuning strategy based on robust performance number for multiple-input multiple-output (MIMO) MPC. However, the above tuning strategies may be not directly used for tuning the large-scale two-dimensional industrial CD MPC, especially in the spatial domain.
The procedure for implementing a paper machine CD MPC control system is shown in FIG. 1. The third step where prediction horizons and optimization weights are selected is often ad hoc and typically evaluated via simulations of the closed-loop system. The state of the art is trial and error which quickly becomes overwhelming for large-scale MPC problems. A one-step static optimizer has been proposed to greatly reduce the computation time while accurately predicting the steady-state performance. For CD process, it has been shown that is possible to separately tune the CD controller in the spatial and temporal domain. However, even with the static optimizer, tuning the multivariable CD predictive controller through trial and error is very difficult even for experienced engineers as there are so many (typically more than 10) tuning parameters. Another practical issue is that there is no indication whether the controller is robustly stable for inevitable model uncertainties.